Device and method for displaying full azimuth angle domain image data

ABSTRACT

A device, system, and method for displaying seismic image data may include computing, from a wide-azimuth data set, a discrete data set associated with an image function at a seismic image point. The discrete data set may be mapped onto a continuous curved three-dimensional surface. The mapped data set may be projected onto a continuous planar surface. The projected data may be displayed as a planar disk. A plurality of continuous planar surfaces, each representing a single image point, may be assembled to form a three-dimensional body, representing a seismic gather of image points. The three-dimensional body may be displayed. Other embodiments are described and claimed.

RELATED APPLICATION DATA

This application is a continuation application of U.S. patentapplication Ser. No. 15/600,485, filed on May 19, 2017, which is acontinuation application of U.S. patent application Ser. No. 15/262,124,filed on Sep. 12, 2016, which is a continuation application of U.S.patent application Ser. No. 14/260,778, filed on Apr. 24, 2014, which isa continuation application of U.S. patent application Ser. No.14/046,327, filed on Oct. 4, 2013, which is a continuation applicationof U.S. patent application Ser. No. 12/663,326, filed on Dec. 7, 2009,which is a national phase application of International Application No.PCT/US2008/066041, filed on Jun. 6, 2008, which in turn claims thebenefit of prior U.S. provisional application No. 60/924,972, filed onJun. 7, 2007, all of which are incorporated herein by reference in theirentirety.

FIELD OF THE INVENTION

The present invention relates to the representation and display of imagedata, such as multi-dimensional angle domain seismic data or otherangle-related three dimensional data.

BACKGROUND OF THE INVENTION

A transmitter located on the earth's surface or elsewhere may transmitsignals, such as acoustic waves, compression waves or other energy raysor waves that may travel through subsurface structures. The transmittedsignals may become incident signals that are incident to subsurfacestructures. The incident signals may reflect at various transition zonesor geological discontinuities throughout the subsurface structures. Thereflected signals may include seismic events. Seismic events including,for example, primary (P) waves and shear (S) waves (e.g., transversewaves in which particle motion may be perpendicular to the direction ofpropagation of the wave) may be used to image subsurface geologicalstructures, for example, transition surfaces or geologicaldiscontinuities. A receiver may collect and record data, for example,reflected seismic events.

Surveys may use large numbers of transmitters and receivers to recordsignals across large geophysical regions. Seismic surveyed regions may,for example, extend to several hundred square kilometers. In somesurveys, the distance between transmitters and receivers may be, forexample, about twenty meters, transmitted signals may travel up to aboutten kilometers, and frequencies of transmitted signals may be aboutfifty Hertz. Other values or parameters may be used. Recorded data maybe collected over intervals of time, for example, ten second intervals,and may be digitized every 4 milliseconds, although other parameters arealso possible. For example, the receiver may collect and/or recordseveral tens or hundreds of terabytes of data. Once collected, therecorded data may be stored and/or transmitted to a storage or dataprocessing device such as a memory, server or computing system.

Some seismic acquisition methods, such as multi-azimuth or wide-azimuthdata acquisition methods, may significantly increase the number oftransmitted and received signals used in order to enhance theillumination of reservoirs below complex structures and increase theprecision of geophysical detection. For such methods, single parameters(e.g., pressure or vertical displacement) or multiple parameters (e.g.,pressure and three displacement components) may be recorded. Both Pwaves and S waves may be recorded. Other types of waves and other datamay be recorded. Such methods may increase the amount of data recordedfor imaging subsurface regions. To accommodate the increased amount ofdata, systems that record, process, image, or otherwise use the data mayrequire increased storage size, increased speed for access to inputand/or output devices, and/or high performance computation (HPC)hardware or the like. Such systems may provide computationally and/orpower intensive services.

Exploration of geophysical regions may include imaging the subsurfaceearth, using seismic data recorded from surveying regions, in order tolocate for example hydrocarbon reservoirs. Seismic imaging methods,which may be referred to as seismic migrations, may be classified forexample into two main categories: wave equation migrations and ray-basedKirchhoff migrations. Both types of migrations may be used to generateimages of the subsurface of the earth. Wave equation migrationmechanisms may use numerical solutions to the wave equation toextrapolate the recorded wavefields into the subsurface of the earth. Ateach level of depth, imaging conditions may be applied to the incidentand reflected wavefields. Ray-based Kirchhoff migrations may beperformed in two stages: ray tracing and imaging. Ray tracing may modelthe propagation of waves (e.g., rays), for example, in a direction froma surface towards an image point in a subsurface region, and/or in adirection from an image point in a subsurface region towards a surface.Ray attributes, such as traveltimes, ray trajectories, slowness vectors,amplitude and phase factors, may be computed along the traced rays. Inthe imaging stage, the ray attributes may be used to obtain an image ofthe earth's subsurface from the recorded seismic data.

Both wave equation and ray-based Kirchhoff migrations may generatecommon image gathers (CIGs). CIGs may include multiple image traces at agiven lateral location. Each image trace may be generated using aportion of the recorded data that has a common geometrical attribute.For example, an offset domain common image gather (ODCIG) may includemultiple image traces, where each trace may be constructed using seismicdata points with the same offset or distance between a source andreceiver on the earth's surface. An angle domain common image gather(ADCIG) may include multiple image traces, where each trace may beconstructed using seismic data points with the same opening anglebetween the incident and reflected rays at the reflection point.

CIGs generated using traces that share a single azimuth may imagegeophysical structures with insufficient accuracy. For example,anisotropy effects show that images obtained from different azimuthangles may be significantly different. Imaging geophysical structures,such as faults, small vertical displacements, and sub-seismic scalefractures (e.g., fractures measuring less than tens of meters, which maybe below the resolution for detection of typical receivers or otherdetection instruments), with desired accuracy, may require imaging alongsubstantially each azimuth angle (which may be referred to for exampleas full-azimuth imaging). Wide-azimuth seismic data may be especiallyvaluable for imaging, for example, below salt dome or salt ladenstructures, such as those in the Gulf of Mexico. Imaging geophysicalstructures using, for example, three-dimensional (multi-azimuth) CIGs,instead of commonly used two-dimensional (e.g., single or narrowazimuth) CIGs, may improve image accuracy and provide additionalinformation about the structures. For example, three-dimensional ODCIGsmay include multiple image traces that have substantially differentazimuth angles on the earth's surface, in addition to substantiallydifferent source-receiver offsets. The offset may be a two-dimensionalvector, for example, having values for in-line and cross-linecomponents, or a length and an azimuth. Similarly, three-dimensionalADCIGs may include multiple image traces that have substantiallydifferent opening azimuth angles at the reflecting surface, in additionto substantially different opening angles. The opening angle may be,e.g., an angle between the incident and reflected rays, measured at thereflection point corresponding thereto. The opening azimuth angle maybe, e.g., the azimuth of the normal to a plane that passes through theincident and the reflected rays. Other angles may alternatively be used.Although three-dimensional CIGs may increase imaging accuracy, they mayalso increase the computational complexity of imaging, visualization,and/or interpretation systems using such gathers. Operation ofthree-dimensional CIGs may also require extensive memory and storagecapacity.

CIGs may be used, for example, in the kinematic and dynamic analysis ofsubsurface structures. For example, kinematic analysis may be used tobuild and update geophysical models using tomography mechanisms.Tomography mechanisms may be used to find a set of model parameters thatsubstantially minimize travel time errors along specular rays (e.g.,ray-pairs that obey principles of Snell's law at the reflectingsurfaces). The travel time errors may for example be measured from thedifferences between locations of the reflection events along the CIGs.Substantially each reflection event within a given CIG may be related toa specific depth. If a “true” reflector (e.g., a reflection surfaceelement) is located at a definite depth and the model parameters are“correct”, then the reflector elements are typically at the same depthirrespectively of the reflection angle or the offset indicated by thespecific trace. When reflection events are not located at substantiallythe same depth (e.g., when reflection events along the CIGs are notsubstantially flat), the measured or picked differences between thereflection depths of different reflection events may be used to estimatethe travel time errors along the specular rays associated with eachtrace. A model may be substantially correct when the seismic reflectionevents along the CIGs are substantially horizontally flat. In order toobtain an accurate model, for example, using an anisotropy modelrepresentation, specular rays and the corresponding travel time errorsfrom varying opening angles (or e.g., offsets) for example, fromsubstantially all azimuths may be used. In some embodiments, suchthree-dimensional CIGs may provide information about the azimuthaldependent travel time errors.

Dynamic analysis may include determining physical and/or materialparameters or properties of target subsurface structures using changesin the amplitude and phase of reflected signals measured, for example,along the CIGs. Multi-azimuth CIGs may make it possible to performazimuthal analysis of amplitude variations with respect to the openingangle (or e.g., offset), which may result in an accurate reconstructionof anisotropy parameters and small scale fractures.

Imaging other than seismic or subsurface imaging for the exploration andproduction of oil and gas, such as for example, shallow seismic imagingfor environmental studies, archeology and construction engineering, maybe performed. These other methods may similarly generate large amountsof data and have large computational needs. Other types of imaging, suchas medical imaging, may also use a relatively large number oftransmitters and detectors and therefore may also use a relatively largeamount of data, which may require large storage and intensivecomputational efforts.

The prior patent application Ser. No. 11/798,996, describes efficientuse, storage, processing, imaging, analysis, visualization andinterpretation of the rich azimuth data in reduced dimensionalcoordinate system. In some other imaging applications, the rich azimuthdata are decomposed into few (e.g., up to eight) azimuthal sectors. Aneed exists for displaying the discrete data, stored for example inreduced dimensional coordinate system, or the azimuthally sectorizeddata, in a continuous full dimensional coordinate system.

SUMMARY

Embodiments of the invention may include computing, from a wide-azimuthdata set, a discrete data set associated with an image function at aseismic image point. The discrete data set may be mapped onto acontinuous curved three-dimensional surface. The mapped data set may beprojected onto a planar surface. A plurality of continuous planarsurfaces, each representing a single image point, may be assembled toform a three-dimensional body, representing a seismic gather of imagepoints. The three-dimensional body may be displayed. Other embodimentsare described and claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The principles and operation of the system, apparatus, and methodaccording to embodiments of the present invention may be betterunderstood with reference to the drawings and the following description,it being understood that these drawings are given for illustrativepurposes only and are not meant to be limiting.

FIG. 1 is a schematic illustration of a spherical display of reflectionangle data associated with a single image point, according to anembodiment of the invention;

FIG. 2 is a schematic illustration of a spherical display of thedirectional data, associated with a single image point, according to anembodiment of the invention;

FIG. 3 is a schematic illustration of a cylindrical display ofreflection angle data associated with a plurality of image points,according to an embodiment of the invention;

FIG. 4 is a schematic illustration of a cylindrical display ofdirectional data associated with a plurality of image points, accordingto an embodiment of the invention;

FIGS. 5 and 6 are flowcharts of methods, according to embodiments of theinvention;

FIG. 7 is a schematic illustration of a system, according to anembodiment of the invention;

FIGS. 8A, 8B, 8C, 8D, and 8E, are schematic illustrations of an imagefunction defined on nodes arranged in various configurations, and mapstherebetween, according to an embodiment of the invention; and

FIGS. 9A, 9B, and 9C, are schematic illustrations of stages ofconstructing a computational mesh, according to an embodiment of theinvention.

For simplicity and clarity of illustration, elements shown in thedrawings have not necessarily been drawn to scale. For example, thedimensions of some of the elements may be exaggerated relative to otherelements for clarity. Further, where considered appropriate, referencenumerals may be repeated among the drawings to indicate corresponding oranalogous elements throughout the serial views.

DETAILED DESCRIPTION OF THE INVENTION Introduction

In the following description, various aspects of the present inventionwill be described. For purposes of explanation, specific configurationsand details are set forth in order to provide a thorough understandingof the present invention. However, it will also be apparent to oneskilled in the art that the present invention may be practiced withoutthe specific details presented herein. Furthermore, well known featuresmay be omitted or simplified in order not to obscure the presentinvention. Unless specifically stated otherwise, as apparent from thefollowing discussions, it is appreciated that throughout thespecification discussions, utilizing terms such as “processing,”“computing,” “calculating,” “determining,” or the like, refer to theaction and/or processes of a computer or computing system, or similarelectronic computing device, that manipulates and/or transforms datarepresented as physical, such as electronic, quantities within thecomputing system's registers and/or memories into other data similarlyrepresented as physical quantities within the computing system'smemories, registers or other, such as information storage, transmissionor display devices. The term “display” may be used herein to describe avisual representation and/or a device for depicting such arepresentation and/or a method or an algorithm for such representation.In addition, the term “plurality” may be used throughout thespecification to describe two or more components, devices, elements,parameters and the like.

Seismic data may include or represent seismic events (or e.g., signals)that reflect and/or diffract at discontinuous objects and/or continuoushorizons. Continuous horizons may include, for example, interfacesbetween geological layers. Discontinuous objects may include, forexample, small scale diffractors, faults, or small scale fractures.

Seismic data collected and/or calculated for seismic imaging may becomplicated and rich with information. For example, seismic data mayinclude multiple components, e.g., traveltimes, offsets, zenith angles,azimuth angles, reflection angles, directions, etc. Each image point maybe computed or defined using multiple components. Each image point mayhave a value for each of the multiple components. However, it may beimpractical to display all of the component values for all of thesecomponents at once to a user. For example, if each component isrepresented in a distinct dimension of a three dimensional (3D) display,at most three of these components may be displayed at once to a user, orthe multiple component values may be used to create the three componentsof the 3D display, and thus the actual source component values may notbe displayed. A standard display may be any a visual representation ofthe spatial properties of the physical universe. In one embodiment, astandard 3D display may represent a geophysical space using, forexample, three components per data point, such as, two Cartesiancomponents of the data point location and a function value at eachpoint. In this embodiment, the image function ƒ may be presented as acurved 3D surface in the Cartesian space, ƒ=ƒ(x, z), where x may be thelateral coordinate and z may be depth. In another embodiment, all threeof the components for each data point of a 3D standard display may beused to specify a spatial location of the data point. Each functionvalue ƒ(x, y, z), where x and y may be two lateral coordinates and z maybe depth, at each point may alternatively be represented by one of aplurality of colors or intensities. In each of these embodiments, astandard display typically shows only a subset of the multiplecomponents to provide a general or simplified overview of a geophysicalarea being studied.

A user may want to view additional or other components or informationnot typically shown in a standard display. For example, a user may wantto view reflection angles and/or directional data at each point, or at aselected point or set of points. A pair of reflection angle anddirectional displays for a point, where the directional display has asingle or narrow range of polar angles and the reflection display hasmultiple or a wide range of polar angles, may indicate that the imagepoint lies on a fault line. Thus, it may be desirable to concurrentlyview reflection angle and directional data corresponding to an imagepoint. These methods may be used for geophysical exploration usingreflection angle gathers and directional gathers. Other features may bedetermined by values of certain components.

Embodiments of the present invention provide a mechanism for displayinginformation different from that displayed with the standard displayusing various types of displays. For example, a “spherical” display(e.g., FIGS. 1 and 2) and a “cylindrical” display (e.g., FIGS. 3 and 4)may be used. Shapes other than spherical and cylindrical displays may beused. One embodiment of the spherical display may display the additionaldata for a single point while one embodiment of the cylindrical displaymay display the additional data for a plurality of points.

A display may represent a function, ƒ. The function may be defined forone or more arguments, e.g., zenith and azimuth. The function may have avalue for the arguments, e.g., typically defined as ƒ(θ,φ), where θ maybe the zenith angle and φ may be the azimuth angle. For example, aspherical display may represent a function (e.g., vs. a reflection anglein FIG. 1; and vs. a direction in FIG. 2) value corresponding to eachcoordinate (e.g., different arguments, zenith and azimuth) of thesphere. The whole spherical display may correspond to a single point ona standard display. Thus, a spherical display may represent the function(e.g., vs. a reflection angle in FIG. 1; and vs. a direction in FIG. 2)value for all polar angle values associated with a single point of thestandard display. The function value at all polar angles may beimportant information when using wide or full azimuth seismic data(e.g., data collected by imaging at many azimuth angles). The sphericaldisplay may represent other image functions, such as, reflectivity,seismic amplitude, or traveltimes. The type of additional data displayedon the sphere may be predetermined, programmed, and/or selected ormodified by a user.

The cylindrical display typically represents a plurality (e.g., a linethat may be vertical, tilted, curved, etc.) of points (e.g., a gather)of the standard display. The axial cross-section of the cylindricaldisplay may represent a meridian gather, which may be a data setrepresenting a plurality of image points having different depths orzenith angles, and the same azimuth angle.

In one embodiment, a plurality of spherical displays, each representinga single point of the standard display, may be combined to concurrentlydisplay a plurality (e.g., a line that may be vertical, tilted, curved,etc.) of such points. In one embodiment, each of the plurality ofspherical displays may be flattened or projected onto a two-dimensional(2D) planar surface forming a planar (e.g., circular) disk. In oneembodiment, the planar disks bounded by 2D curved lines may be stackedor otherwise assembled or combined to form a cylinder (e.g., regular,tilted, with a curved axis, etc.) representing the plurality (e.g., avertical line, a tilted line, a curve, etc.) of points of a standardmodel or representation of the geometry of the physical universe. Forexample, the plurality of points shown in a standard model maycorrespond to an image gather or other object.

According to this construction, each point of a standard model maycorrespond to a separate spherical display. Each spherical display mayin turn correspond to a planar disk or other planar figure (e.g., theflattened sphere). Each planar disk may in turn correspond to a point onthe axis of a cylinder (e.g., as a normal cross-section of the cylinderat the point), constructed by stacking the planar disks.

A user may manipulate the view of the spherical or cylindrical displays.For example, a user may move a cursor or mouse or other pointing orinput device to select or search along a length (e.g., an axis ofsymmetry) of the cylinder display. For example, when the user selects apoint or location on the cylinder (e.g., along an axis of symmetry),then the cross-section (e.g., planar curve), spherical display, and/orpoint of the standard model, corresponding thereto, may be displayed.The user may use the data to identify features such as subterraneanfaults. For example, a user may select to scan image points, to view anddisplay pairs of directional and reflection angle (e.g., or offset)displays corresponding to the selected image points. The user may searchfor pairs of displays, where the directional display has a single ornarrow range of polar angles and the reflection display has multiple ora wide range of polar angles. Such pairs of displays may indicate, forexample, that the corresponding image point is located on a fault line.

A polar angle may be, e.g., a two-dimensional vector (e.g., defined by azenith and an azimuth angle). A zenith is, e.g., the angle between theradius-vector (connecting the center of the sphere to that point) andthe polar axis of the sphere. An azimuth angle is, e.g., an anglebetween a reference direction in the equatorial plane and a projectionof the radius-vector on the equatorial plane. Other angles, directions,orientations, relationships therebetween, and/or definitions thereof,may alternatively be used.

Each of the cylindrical, spherical, and standard displays mayindividually show different information corresponding to the samegeophysical data or subsurface space. In some embodiments, when thecylindrical, spherical, and standard displays are simultaneouslydisplayed (e.g., adjacently) the information of each may be compared.For example, while the user scans a column with a cursor, acorresponding indicator may scan a standard display to indicate to theuser which geophysical location is being selected.

In another embodiment, the user may select (e.g., by clicking orhighlighting) a geophysical location of the standard display to bedisplayed as a sphere or cylinder. The user may operate an input device(e.g., a mouse or keyboard) to manipulate, select, highlight, orotherwise indicate a seismic image point or a plurality of seismic imagepoints corresponding to a line in a physical space. In response to suchan indication, a display may display (e.g., on a graphical userinterface) a representation of the indicated data to a user. Forexample, in one embodiment, a user may click or otherwise indicate acoordinate or point of the standard display and a correspondingspherical or other display may appear or “pop-up” (e.g., adjacentthereto). Likewise, a user may select (e.g., by dragging a cursor) aline or other plurality of points and a corresponding cylindricaldisplay may pop-up (e.g., adjacent thereto). In a system having morethan one monitor, a standard display may be displayed on one monitor,and details with specific data may appear on a second monitor or in anew separate window of the same monitor.

In other embodiments the displays may be rotated, translated, shifted,sliced, bent, rescaled, colored, zoomed and/or otherwise moved orreshaped.

It may be appreciated by one skilled in the art that althoughembodiments of the invention are described in terms of sphere orspherical display, other shapes may be used, e.g., ellipsoids, torus,hyperboloids, polyhedron, which may be, symmetric or asymmetric, andregular or irregular. It may be appreciated by those skilled in the artthat although embodiments of the invention are described in terms ofplanar disks or flattened sphere, other shapes of 2D figures may beused, e.g., ellipses and other types of conical sections, polygons, orapproximations thereof, etc. It may be appreciated by those skilled inthe art that although embodiments of the invention are described interms of cylinders, other shapes may be used, e.g., cones, prisms,pyramids, polyhedrons, etc., and/or geometric shapes having a line, or aplane of symmetry.

Reference is made to FIGS. 8A, 8B, 8C, 8D, and 8E, which are schematicillustrations of an image function defined on nodes arranged in variousconfigurations, and maps therebetween, according to an embodiment of theinvention. The specific figures and data points in each of FIGS. 8A-8Eand the relationships therebetween are not limiting. It may beappreciated by those skilled in the art that these figures and theelements thereof are only examples and that other mechanisms,structures, relationships, mathematics, and abstractions, may be used toprovide embodiments of the invention.

FIG. 8A shows an irregular discrete data set 815 representing the imagefunction as defined on irregular nodes 810. FIG. 8B shows a regulardiscrete data set 825 representing the image function as defined onregular nodes 820 of a computational mesh 822. FIG. 8C shows acontinuous curved three-dimensional surface 835 representing the imagefunction as defined on spiraling nodes 830. FIG. 8D shows a curvedplanar surface 845, representing the image function as defined on planarnodes 840. The image function may be represented on the planar surface845 by a two-dimensional region with a curved boundary (e.g., such asregions 365 and 465 of FIGS. 3 and 4, respectively). FIG. 8E shows athree-dimensional body 855 representing the image function as defined onnodes 850 of a cylindrical body. The irregular discrete data set 815,regular discrete data set 825, continuous curved three-dimensionalsurface 835, planar surface 845, and three-dimensional body 855 may bethe values of the image function at the irregular nodes 810, regularnodes 820, spherical spiral nodes 830, planar curve nodes 840, and nodes850 of the cylindrical body, respectively.

The image function may be any additional or other components orinformation. For example, the image function may be defined vs. areflection angle, vs. offset, and/or directional data function. Thevalue of the image function may be expressed, for example, as a colorvalue on a color map.

The locations of the nodes of FIGS. 8C, 8D, and 8E typically do notcorrespond to the physical locations of the image points to which theycorrespond. For example, a node located to the relative “right” on thecontinuous curved three-dimensional surface 835 of FIG. 8C maycorrespond to a greater azimuth angle and not necessarily to a moreeastern physical location of a subsurface or geophysical region. Forexample, a plurality of data points in FIGS. 8A and 8B describe aplurality of directions for a single physical point in a 3D space. Thesedirections are arranged irregularly in FIG. 8A and regularly in FIG. 8B.“Regularly” arranged nodes may indicate that a horizontal line in FIG.8B represents, e.g., a latitude line, with a fixed zenith angle andvariable azimuth. “Regularly” arranged nodes may likewise indicate thata vertical line in FIG. 8B represents, e.g., a meridian line or a subsetof the meridian gather, with a fixed azimuth and variable zenith,related to a single physical point. Other arrangements of nodes may beconsidered “regular”. For example, an alternatively regular arrangementof nodes may include nodes located at the vertices of a regularpolyhedron (e.g., an icosahedron, which has 12 vertices and 20 faces)inscribed into a sphere. The vertices of the regular polyhedron may bethe primary nodes of a regular grid. The regular grid may be refined by,e.g., splitting the faces of the polyhedron into regular (e.g., but notnecessarily equal) pieces for generating other (e.g., additional) nodesof the regular grid. This alternate embodiment is, for example,described in further detail in reference to FIGS. 9A, 9B, and 9C.

Each of FIGS. 8A, 8B, 8C, and 8D typically represents the image functiondata corresponding to a single point (not shown) in a physical space andFIG. 8E typically represents the image function data corresponding to aplurality of image points (e.g., a line 880) in a physical space. Forexample, three-dimensional body 855 may include a plurality of planarsurface 845, each representing an image point on a line of a physicalspace. Planar surface 845 may have a curved two-dimensional line orregion for representing the image function at the image point. Forexample, the center points of each of the planar surface 845 may bestacked (e.g., along the line 880) according to the locations of theircorresponding image point on the line of the physical space.

A first map 860 may be used to map input data from the irregulardiscrete data set 815 to the regular discrete data set 825. The firstmap 860 may be used to normalize or regularize irregular data. The firstmap 860 need not be used when the input data is already regular (e.g.,defined at regularly spaced nodes of a coordinate system, where eachnode represents a definite polar angle, i.e., a fixed direction inthree-dimensional space).

A second map 865 may be used to map input data from the regular discretedata set 825 to the continuous curved three-dimensional surface 835.Using the regular discrete data set 825, the second map 865 may be usedto generate a continuous distribution of the image function for a singleimage point on the continuous curved three-dimensional surface 835. Datamapped to the continuous curved three-dimensional surface 835 may berepresented on for example the spherical displays 100 and 200 of FIGS. 1and 2.

Operating on input data using the first and second maps 860 and 865 inorder may be referred to as “gridding”. Operating on input data usingthe second map 865 but not the first map 860 may be referred to as“interpolation”. Therefore, interpolation may be a sub-operation ofgridding. First and second maps 860 and 865 are described in furtherdetail herein, e.g., in the section titled, “Spherical Gridding”. Theconstruction of the nodes 820, which may be collectively referred to asa “computational mesh” 822, is described in further detail herein, e.g.,in the section titled, “Computational Mesh” and in reference to FIGS.9A, 9B, and 9C. FIGS. 9A, 9B and 9C describe a mechanism for generatinga regular mesh (e.g., computational mesh 822 of FIG. 8B). The regularmesh may be generated on a sphere. The mesh described in 9A, 9B, and 9Chas cells of approximately equal area, or at least, cells of similarareas. The mesh preferably does not have singularities at the poles ofthe sphere.

A third map 870 may be used to map input data from the continuous curvedthree-dimensional surface 835 to the curved two-dimensional surface ofthe planar surface 845. The third map 870 may be, for example, aprojection map, of the continuous curved three-dimensional surface 835or any other map for flattening or transforming or expanding thecontinuous curved three-dimensional surface 835 to the planar surface845. A projection map may describe a surjective or “onto” map as areknown in the art. A projection map may describe a function for mappingdata from a first coordinate space (e.g., N-dimensional) to a second(e.g., N-1 dimensional) coordinate space. Alternately, third map 870 maybe, for example, an expansion map. The expansion map may project athree-dimensional surface into a two-dimensional surface by “unraveling”or “unfolding” the three-dimensional surface. The third map may includeother or additional projections, such as, e.g., cylindrical,pseudo-cylindrical, hybrid, conical, pseudo-conical, azimuthal(projections onto a plane), conformal, equal-area, equidistant,gnomonic, retro-azimuthal, compromise projections, or the like.

A fourth map 875 may be used to map input data from a set of the planarsurfaces 845 to the three-dimensional body 855. The fourth map 875 maybe used to generate a continuous distribution of the image function fora plurality of image points (e.g., corresponding to a line in a physical3D space) on the three-dimensional body 855. Data mapped to thethree-dimensional body 855 may be represented on the cylindricaldisplays 300 and 400 of FIGS. 3 and 4.

Embodiments of the invention include operating on a discrete data set(e.g., irregular or regular discrete data sets 815 and 825,respectively) representing a single or a plurality of image points of aphysical 3D space with a sequence of maps in a predetermined order(e.g., first map 860, second map 865, third map 870, and/or fourth map875) to generate a continuous curved three-dimensional surface 835(e.g., a sphere) and a three-dimensional body 855 (e.g., a cylinder),respectively.

It may be appreciated by those skilled in the art that maps describedherein are only one example and that other maps, functions,transformations, or relationships may be used to map data between thevarious nodal configurations, e.g., irregular nodes 810, regular nodes820, spiraling nodes 830, planar nodes 840, and/or cylindrical nodes850. It may be appreciated by those skilled in the art that any map(s)may be combined or separated into other and/or different numbers of mapsto perform equivalent operation(s). It may be appreciated by thoseskilled in the art that nodes and the configurations thereof describedherein are only one example and that configurations or arrangements ornodes other than irregular nodes 810, regular nodes 820, sphericalspiral nodes 830, planar surface nodes 840, and/or cylindrical nodes 850may be used to represent equivalent information.

Embodiments of the invention provide a system and methods for displayingcontinuous full-azimuth angle domain image data and/or wide-offsetdomain image data at a given image point of the discrete data set suchas data set 815 of FIG. 8 (e.g., on the spherical displays 100 and 200of FIGS. 1 and 2) or at a set of image points (e.g., on the cylindricaldisplays 300 and 400 of FIGS. 3 and 4). The full-azimuth angle and/orwide-offset domain image data of the discrete data sets 815 and 825 mayinclude, for example, angle domain common image gathers (ADCIG) andoffset domain common image gathers (ODCIG), respectively. Embodiments ofthe invention include accepting the (e.g., azimuthally) discrete dataset, mapping the discrete sets of data to the continuous curvedthree-dimensional surface 835, for example, creating three-dimensionalADCIGs in the directional or reflection angle domains. Embodiments ofthe invention include, for each image point, using, for example,spherical gridding mechanism or other methods or techniques forinterpolating and extrapolating data onto a spherical or otherwisecurved surface. It may be appreciated by those skilled in the art thattechniques other than spherical gridding (e.g., the first and secondmappings 860 and 865 in FIG. 8) or interpolation (e.g., only the secondmap 865) may be used for generating spherical, cylindrical, or otherwiseshaped displays for representing wide-azimuth data, as described herein.Embodiments of the invention include projecting the data represented onthe curved surface to a planar surface. For a gather of image points(e.g., an ADCIG), embodiments of the invention include combining thedata sets associated with several planar surfaces into athree-dimensional body, such as a circular cylinder or other cylinder.It may be appreciated by those skilled in the art that shapes other thanspheres and cylinders may be used.

Embodiments of the invention include a mechanism for mapping angledependent image point data onto a spherical or otherwise curved surface,using, for example, a spherical gridding, interpolation/extrapolation orother methods. Embodiments of the invention include displaying angledependent image point data on a spherical or otherwise curved surface.The image function data (e.g., the reflectivity) may be displayed as afunction of the direction angles, reflection angles, and/or offsets.Embodiments of the invention include projecting or expanding the datadefined on a curved surface to the data defined on a planar surface.Embodiments of the invention also include cylindrical displays fordisplaying simultaneously image data for a plurality, for example, agather of image points (e.g., a set of image points located along avertical line at a given horizontal location). As described in thereferred patent U.S. patent application Ser. No. 11/798,996,full-azimuth angle gathers (e.g., ADCIGs) may be used to generate aspherical spiral geometry. These types of gathers may be referred to as“Spiral ADCIGs”, where a directional gather may be referred to as a“Spiral-D” and a reflection gather may be referred to as a “Spiral-R”.

It may be appreciated by those skilled in the art that embodiments ofthe present invention may be applied to any seismic processing andimaging system. Embodiments of the present invention may be used forgenerating displays and visualizations in various areas or fields, suchas, for example, exploration and production of oil and gas, imaging ofthe shallow earth model for environmental study (e.g., using datacollected using seismic and/or ground penetration radar (GPR) methods),construction engineering (e.g., to identify locations of pipes),construction safety and security (e.g., to identify holes and channels),medical imaging (e.g., using computed tomography (CT), magneticresonance imaging (MRI), and ultra-sound devices), non-destructivematerial inspection, inspection of internal items for security reasons(e.g., homeland security), marine sonar, and antenna and radar systems.

The Proposed Displays

Reference is made to FIGS. 1-4, which are schematic illustrations ofdisplays or visualizations of continuous data according to embodimentsof the invention. FIGS. 1 and 2 are spherical displays related to aspecific image point, according to one embodiment. FIGS. 3 and 4 arecylindrical displays related to a specific gather of image points,according to one embodiment. The data displayed in FIGS. 1-4 may be dataadditional to that typically shown in standard displays. For example,these displays may be generated at the request of a user to have a“closer look” or additional information associated with one or more datapoints of a geophysical region. The additional information may includefor example an image function of reflection angle, offset, anddirectional data for the one or more data points.

Displays 100, 200, 300, and 400 may be generated for example by themapping of data as shown in FIGS. 8A, 8B, 8C, 8D, and 8E, although othermethods of creating these displays may be used. The input data for thedisplays may be discrete data set 815 or 825, for example, of an imagefunction defined at a plurality of discrete nodes 810 and 820 of FIG. 8(e.g., referred to as “control points”). The location of nodes 810 and820 may be defined by a discrete geometry. The nodes 810 and 820 may beirregularly and regularly spaced, respectively, with respect to acoordinate system (e.g., a spiral coordinate system).

Embodiments of the invention may include a gridding mechanism (e.g.,using first map 860 of FIG. 8) for normalizing the irregular discretedata set 815 to generate the regular discrete data set 825. The regulardiscrete data set 825 may be interpolated (e.g., using second map 865)to generate the continuous curved three-dimensional surface 835. Thecontinuous curved three-dimensional surface 835 may represent the imagefunction at a single image point.

In one embodiment, the discrete data set 825 of FIG. 8 may be mapped ortransformed by second map 865 to a continuous surface by simulating theelastic bending of a thin shell reinforced by “springs” or elasticstructures. In one embodiment, “springs” may be data abstractionssupporting the nodes of the shell at discrete control points. Forexample, the “springs” may be, contracted, or expanded, to shift,“squeeze”, or reorient, the nodes 810 and/or 820 in order to provide abetter coincidence of the deformed shell with the data values at thereference nodes, such as, the spiraling nodes 830 of the continuouscurved three-dimensional surface 835, as described herein. It may beappreciated by those skilled in the art that techniques other thanspherical gridding and/or interpolation may be used for generating acontinuous image function, as described herein.

In some embodiments, exact accuracy is not required for a griddingmechanism. Instead, a gridding mechanism may be adjusted for achieving abalance between accuracy and a continuity of gridding. “Springs” may beused to provide a better continuity of gridding. It may be appreciatedthat the “springs” and the use and elasticity thereof, as describedherein, are data abstractions and computational abstractions, and may berepresented by comparable software programs or sets of instructions,mathematical equations, data value, and/or visual representations.

In some embodiments, when control points or nodes 810 and/or 820 areshifted toward the in-line or cross-line directions, there may an optionto “squeeze” the computational mesh 822 (e.g., and corresponding orspherical displays 100 or 200 of FIGS. 1 and 2), for mapping thediscrete nodes 810 and/or 820 to the continuous curved three-dimensionalsurface 835 that better fits the real image data, such as, a spheroid ora general ellipsoid with all three axes distinct (e.g., scaleneellipsoid).

In some embodiments, “squeezing” may include transforming, distorting,refitting, resizing, reshaping, or other manipulation effects to alter ashape of the continuous curved three-dimensional surface 835 and/orcomputational mesh 822 (e.g., as described herein) used to generate theshape.

In one embodiment, the discrete data set may be defined on discretenodes of a spiraling geometry, for example, as described in U.S. patentapplication Ser. No. 11/798,996. For example, with a spherical spiraldiscretization, the location of input points may be described by asingle parameter, e.g., the normalized area swept by spiral coils or thenormalized arc length of spiral. The display mechanisms may includespherical gridding or other interpolation or extrapolation techniques,such as for example, local spherical interpolation, or Fourier transformon a spherical surface. These mechanisms may map an image functiondefined at discrete regular or irregular nodes 810 and/or 820 onto aspherical or otherwise continuous curved three-dimensional surface 835,with a continuous distribution of the image function defined through acorresponding (e.g., “squeezed”) curved surface.

The gridding mechanisms described herein are non-limiting examples ofmany possible methods for transforming or a mapping a discrete data set815 and/or 825 to a continuous curved three-dimensional surface 835.

Other gridding mechanisms, maps, and/or transformations, may be used.

Spherical Gridding

Objective of Spherical Gridding

“Normalization” may refer to for example mapping irregular data to aregular data set (e.g., using first map 860). “Interpolation” may referto for example, using discrete data to generate a continuousdistribution of an image function (e.g., using second map 865).“Gridding” may refer to for example the combined steps of normalizationand interpolation, for example, executed in that order.

Embodiments of the invention may include interpolating or mapping adiscrete data set such as data sets 815 or 825 of FIG. 8 of an imagefunction to a continuous image function on a continuous curvedthree-dimensional surface such as surface 835 (e.g., using first and/orsecond maps 860 and 865).

The discretely defined image function may be defined on an input grid ofnodes 810 or 820. When the nodes 810 of the input grid are irregular, astandardized or uniform grid may be used to regularize or normalize theinput grid to generate an output grid of nodes 820 defining a continuousimage function. For example, the data in the input grid that differsfrom the uniform grid may be adjusted to fit (e.g., using first map860). The input grid may be for example a spiral coordinate system. Theoutput grid may be a conventional spherical mesh with constantresolutions in zenith and azimuth, or any other conventionalvisualization mesh on a spherical surface or other curved surface. Othergrids may be used.

Embodiments of the invention include a gridding mechanism to interpolatean image function at a point onto a continuous spherical, ellipsoidal,or otherwise curved three-dimensional surface 835 (e.g., using secondmap 865). The curved three-dimensional surface 835 may be displayed asspherical displays 100 and 200 representing the reflection angle imagefunction and the directional data, respectively, in FIGS. 1 and 2, for asingle point.

The interpolated image functions for each point may be flattened orprojected to the curved two-dimensional surface of planar surface 845(e.g., using third map 870), for a single point.

A plurality of curved two-dimensional planar surfaces 845, eachrepresenting the image function at a point, may be stacked along theline 880 to form a cylindrical or other three-dimensional body 855(e.g., using fourth map 875), representing the image function at aplurality of points. The three-dimensional body 855 may be displayed asdisplays such as cylindrical displays 300 and 400 representing thereflection angle image function and the directional data, respectivelyin FIGS. 3 and 4, for a plurality of points. The position at which eachof planar surfaces 845 is stacked along the line 880 may, for example,correspond to the relative spatial arrangement of the image pointrepresented thereby. Thus, a user may scan the line 880 in a direction(e.g., “upward”) to display cross-sections of the cylindrical display(e.g., as planar surfaces 845 or corresponding three-dimensionalsurfaces 835 unflattened by an inverse map of 870) of the image functionat different image points of the physical universe in a corresponding(e.g., “upward”) direction.

The initial image function may be defined at discrete control points(e.g., irregular or regular nodes 810 or 820). Embodiments of theinvention provide first and/or second map 860 and/or 865 to generate acontinuous distribution of the image function on a curvedthree-dimensional surface 835 from input points of the discrete data set815 or 825. The components of the polar angles (e.g., zenith andazimuth) of the input (e.g., control) points may be presented in, e.g.,a Cartesian coordinate system or a spherical coordinate system (e.g., atthe nodes of a spherical or ellipsoidal spiral), or other coordinatesystems. The input points may have any locations, may be regularly orirregularly spaced, and may be arranged in any suitable order orsequence. The output function may be a continuous image function definedat any point or at all points of the spherical or ellipsoidal surface,or at all points of a part of the surface (e.g., such as a sphericalhemisphere or “ellipsoidal cap”). It may be appreciated that differentpositions on the curved three-dimensional surface represent a singlephysical point in a 3D space and a plurality of directions correspondingto the point. At the control points (e.g., irregular nodes 810), theoutput values may coincide or nearly coincide with the input data. Insome embodiments, an adjusting parameter (e.g., such as weights of thedata at the control points) may be applied to the interpolation map(e.g., as part of the first map 860) for balancing the benefits ofcontinuity (e.g., smoothness) of the distribution and accuracy of thesolution to the image function at the control points (e.g., achieving abest fit for matching the input and output data of the interpolationfunction). In some embodiments, the output image function may begenerated to exactly match the control or input data. Alternatively, theoutput image function need not exactly match the input image function.In such embodiments, when the exact fit requirement is relaxed, theoutput image function may be a more smooth and continuous distributionof the mapped function through the curved surface with smallergradients.

Gridding may include simulating the elastic bending of a thin shell orcomputational mesh 822 of FIG. 8B, reinforced by springs. This model isonly one of a variety of possible approaches, and other models andapproaches may be used. It may be appreciated by those skilled in theart that the shell and the springs are abstractions, and that there isno real physical structure that comprises shell and springs. The springabstraction may be located at the nodes of control points. Typically, anexact match between the interpolated (e.g., output) function and theinput function at the control points is not required. The sphericalgridding method may be used to interpolate image data, for example, whenthe angle domain is not fully illuminated.

Spherical Gridding in the Local Angle Domain

The system of incident and reflected waves (e.g., or ray pairs) defininga geophysical region may be defined by the directional and reflectionsubsystems of the local angle domain (LAD). The directional system mayinclude two components of polar angles describing the zenith (e.g., dip)and the azimuth angles of the normal to the reflection surface element.The reflection system may include two components describing an openingangle between incident and reflected rays and an opening azimuth oralternatively, an offset magnitude and an offset azimuth, where theoffset is specified on the earth's surface. Together, the directionaland reflection subsystems may define the position of each image point inthe LAD.

For each image point, two angle domain imaging systems may be created,e.g., the directional and reflection subsystems. Other than two angledomain imaging systems may be used. Both imaging systems may be definedon a curved surface, such as, the unit sphere (e.g., the reflectionsubsystem in FIG. 1; the direction subsystem in FIG. 2). A point on thecurved surface may be defined by two components of the polar angle(e.g., the zenith and azimuth). Each point in the direction system onthe spherical surface may correspond to a certain direction of theinward normal to the reflection element, with all possible reflectionangles and their azimuths summed Each point in the reflection system onthe spherical surface may correspond to a certain opening angle andopening azimuth, with all possible directions of the ray pair normalsummed up.

Other subsystems, angles, components, or relationships thereof may beused.

Principles of Spherical Gridding

The term “interpolating” may be used for example to describe estimatingfunction values at (e.g., arbitrary or specific) points between nodes,where the nodes or control points may be evenly or regularly spaced. Theterm “gridding” may be used for example to describe interpolating, wherethe nodes may be irregularly spaced. Thus, gridding may be a moregeneral form of interpolation and interpolation may be considered astage of the gridding procedure. In the interpolation problem, afunction may be defined at evenly (e.g., or at least regularly) spacedgrid nodes of an n-dimensional space. In the gridding problem, the inputpoints may be irregularly spaced (e.g., not necessarily at the gridnodes). In one embodiment of the gridding technique, an input mesh maynot exist and the input control points may be an unordered or randomcollection of points in a finite (e.g., or bounded) space. In thisembodiment, the gridding mechanism may generate a continuousdistribution of a function estimating all points of the finite space.

The gridding procedure may include generating a regular computationalmesh, e.g. 822 (e.g., and the values of an image function estimated atthe spiraling nodes 830) and interpolating the image function betweenmesh nodes 830 for generating a continuous output image function. Theinput image function may be fit exactly, or alternatively,approximately, to the output image function. When the input and outputdata are fit to match less than perfectly, the output data may be fit tobe more continuous (e.g., having fewer and/or less drasticdiscontinuities).

When generating the regular computational mesh 822, the unknowns may bethe nodal values of the image function. Between the nodes, the functionbehavior may be defined by, for example, interpolation polynomials. Tofind the nodal values, the energy of the surface defined by the imagefunction may be minimized, for example, taking into account the controlvalues, either exactly (e.g., to generate the exact output values at thecontrol points), or using additional energy terms (e.g., for greatercontinuity).

When an image function is displayed on a surface such as continuouscurved three-dimensional surface 835, e.g., as a thin elastic shell,shell elastic displacements may be modeled by the image function, andthe energy per unit area (e.g., the specific energy) may be approximatedby the curvature of the shell squared. The surface defined by the imagefunction (e.g., having minimized energy) may be an elastically deformedsurface that initially (e.g., before deformation) had a spherical, anellipsoidal, or another curved form. The surface may be supported bypre-stretched springs at the locations of control data points. In suchembodiments, the input data may be approximately fit. Alternatively,normal displacements may be specified at the control points. In suchembodiments, the input data may be exactly fit.

After the grid values at nodes 820 are established for generating forexample the computational mesh 822, the same interpolation polynomialsmay be used for interpolation to estimate the image function values atarbitrary points between the regularly spaced nodes 820 of thecomputational mesh 822, for example, at nodes 830 of the spiralinggeometry, or at the other nodes. Therefore the stages of interpolation,e.g., generating the computational mesh 822 and interpolating the imagefunction along the mesh, are typically related.

Gridding through a planar or curved surface (spherical, ellipsoidal,etc.) may provide the function values at the spiraling nodes 830 orcontrol points as well as partial derivatives (gradient components) ofthe function at the control points. For an ellipsoidal surface, thepartial derivatives of the function at the control points may be thederivative of the image function with respect to the zenith and having aconstant azimuth, with respect to the azimuth and having a constantzenith, or with respect to the azimuth and having a constant verticalcoordinate (e.g., which may be a different embodiment for a scaleneellipsoid). In addition, the derivative of the input function may bedetermined in any arbitrary direction on a curved surface.

Computational Mesh

The complexity of a computational mesh used with embodiments of theinvention, such as the computational mesh 822, may be defined by forexample the number of recursion levels used to generate the mesh. Themesh may be generated on the unit sphere or any other n-dimensionalspace. When the three-dimensional curved surface is different from theunit sphere (e.g., an ellipsoid) the mesh may be generated on the unitsphere and then formed into another (e.g., ellipsoidal) shape.

Reference is made to FIGS. 9A, 9B, and 9C, which are schematicillustrations of stages of constructing a computational mesh accordingto embodiments of the invention, such as the computational mesh 822,according to an embodiment of the invention. The computational mesh 822(e.g., the generated mesh) is regular, and may or may not be uniform.The specific figures and mechanism described in reference to FIGS. 9A-9Care not limiting. It may be appreciated by those skilled in the art thatthese figures and the elements thereof are only examples and that othermechanisms, structures, relationships, mathematics, and abstractions,may be used to provide embodiments of the invention.

In one exemplary embodiment, the computational mesh 822 of FIG. 8C maybe initially formed from a polyhedron 900 of FIG. 9A, such as, a regularicosahedron with (e.g., 12) vertices 902, (e.g., 20) triangular faces904 and (e.g., 30) edges 906, e.g., inscribed in a sphere 910. Thisgeometry may correspond to the zero recursion level for generating acomputational mesh such as the computational mesh 822. Other shapes,polyhedrons, and numbers may be used.

To obtain the next level of recursion, in FIG. 9B, flat triangular faces904 may be replaced by spherical triangles 912 (e.g., having curved faceand/or edges). For example, points A, B and C are vertices 902 of anarbitrary one of triangular faces 904. Point O is the center of thesphere 910 and lines OA, OB, and OC are radii of the sphere 910. LinesAB, BC, and CA, the edges of the triangular faces 904 of the polyhedron900 (e.g., icosahedron), may be replaced by curved approximations of thelines 918, such as, arcs of “great circles” (or geodesic curves).

To obtain the next level of recursion, in FIG. 9C, each sphericaltriangle ABC 912 may be split by medians into (e.g., four) derivativespherical triangles: (e.g., three) peripheral triangles 916 (of the samearea), labeled TA, TB and TC and (e.g., one) central triangle 914 (ofdifferent area), labeled TM in FIG. 9C. For example, in FIG. 9C newpoints 930, M_(AB), M_(BC) and M_(AC), e.g., the centers of sphericalarcs AB, BC, and AC, respectively, may form additional vertices 902,triangular faces 904, and edges 906 forming a new polyhedron, e.g.,different from polyhedron 900 (e.g., having more vertices, faces, andedges). Each new edge 906 may yield an additional node 930, for example,giving a polyhedron with e.g., 42 nodes (of course other numbers ofnodes may be used).

Such a procedure may be repeated multiple times, e.g., each time thelevel of recursion increasing by one.

For the last (e.g., highest) level of recursion, the spherical triangles912 may be approximated by flat triangular faces 904. The gridding maybe modeled by bending deformations of a three-dimensional spatialelastic shell consisting of the flat triangular faces 904 (e.g.,corresponding to the last level of recursion). As the number ofrecursions increases, the polyhedron 900 converges to approximate thesphere 910 of FIG. 9A. At a certain recursion level (e.g., 4 or 5,although other numbers may be used), the surface of the polyhedron 900may be determined to be sufficiently close to spherical.

The vertices 902 of the polyhedron 900 of FIG. 9A may be the nodes 820of the computational mesh 822 of FIG. 8B. The spherical shell may besimulated to be resting on pre-loaded springs located at the controlpoints. Thus, the spring locations need not necessarily coincide withthe mesh nodes. The direction of the force exerted by a spring may benormal to the spherical or the ellipsoidal surface at the control point.The elastic displacements at the shell nodes may be calculated, forexample, by the Finite Element Method.

The nodes 830 may be numerated globally, for example, in a specificmanner: first the “north pole”, having the highest z coordinate, thenthe nodes of the northern hemisphere having the next z level and other zlevels, then the equator, the levels of the southern hemisphere, andfinally the south pole, having the lowest z coordinate. This numerationmethod may yield a small or minimal band width of a resolving matrix.Other numberings, and other methods of global numbering, may be used.

The whole sphere or ellipsoid need not be analyzed at once. In someembodiments, the input data (e.g., the control points) may be locatedwithin a definite range of zenith angles, for example, that do notexceed a pre-defined maximum zenith value. In such embodiments, only apart of the whole surface (e.g., a spherical or an ellipsoidal cap) maybe analyzed.

In embodiments when the reflection subsystem includes the offset valueand the offset azimuth (e.g., instead of the opening angle and theopening azimuth), gridding may be performed on a flat (e.g., planar)surface (e.g., instead of a spherical surface). Since angle gather datatypically corresponds to different directions at an image point or at aset of image points (e.g., along a vertical line), different directionsat a single point may be represented by a curved surface, such as aspherical surface. In contrast, an offset gather (e.g., or the specificphysical location in depth thereof) typically corresponds to differentlateral shifts between the source and the receiver on the surface of theEarth. These shifts typically have, e.g., two Cartesian components, xand y, and the Earth surface and are represented by a planar or flatsurface.

Northern and southern hemispheres are relative terms describing regionsabove and below an equator (e.g., the widest circumference of anoriented body), respectfully. Similarly, terms such as equator,meridian, south, north, vertical, horizontal, lateral, perpendicular, orother orienting terms are relative terms, depending on a viewer'sperspective or vantage point.

Spherical Display

Reference is again made to FIGS. 1 and 2, which show spherical displays,related to a specific image point, for the reflection and directionalsubsystems, respectively. The output data for the displays 100 and 200(e.g., of continuous curved three-dimensional surface 835 of FIG. 8C)may be specified at the nodes of spherical or ellipsoidal surface (e.g.,spiraling nodes 830 of FIG. 8C). The shape of the displays 100 and 200may be determined by the construction of the computation mesh 822 ofFIG. 8B (e.g., as described in reference to FIGS. 9A, 9B, and 9C). Thedisplays may be for example continuous spherical or ellipsoid surfaces,or portions (e.g., a cap) thereof. In some embodiments, there may be aplurality or range of colors that corresponds to the range of the imagefunction values. The surfaces of the displays may have a color or othervisual indication at each point corresponding to the image functionvalue at that point.

In FIGS. 1 and 2, different colors are represented by differently shaded(e.g., cross-hatched) regions 165 and 265, respectively. The display mayprovide numerical output values (e.g., corresponding to therepresentative color value) at any point of the curved surface, forexample, when the point is selected or highlighted by a user. Thedisplay may provide the input values corresponding to the interpolatedor gridded displayed output values, for example, for comparison betweenthe input and the output values at the control points.

The accuracy of the gridding and the visualization may be defined forexample, by the computational complexity and the visualizationcomplexity. The computational complexity may be the number of recursionlevels of the computational mesh 822. The visualization complexity maybe a similar number, for example, of the sectioning of the graphicsgrid. Typically, these two numbers coincide, or are close. The displaymay be rotated, shifted, zoomed, enlarged, sectioned, inverted, orotherwise transformed or translated.

Cylindrical Display

Reference is again made to FIGS. 3 and 4, which show cylindricaldisplays 300 and 400, each related to a plurality of image points, forthe reflection and directional subsystems, respectively. Each ofcylindrical displays 300 and 400 may be formed as an assemblage ofmultiple planar disks (e.g., curved two-dimensional planar surface 845of FIG. 8D by the fourth map 875). Each planar disk may be formed byprojecting the continuous curved three-dimensional surface 835 (e.g., asphere, ellipsoid, or otherwise 3D curved surface) onto a plane or 2Dspace. Each curved 3D surface may represent a point (e.g., having aphysical location in the Cartesian coordinate system). Thus, each 3Dbody or cylindrical displays 300 and 400 may represent a plurality ofpoints (e.g., each having a unique depth). The cylindrical displays mayrepresent image points defined by coordinates other than depth. In oneembodiment, cylindrical displays 300 and 400 may display substantiallysimultaneously a plurality (for example, a gather) of image points, inan integrated display. For example, each display 300 and 400 maycorrespond to a set of two or more image points with substantially thesame lateral location and substantially different depths.

Consider several directional displays (e.g., FIG. 2) or reflectiondisplays (e.g., FIG. 1) in spherical/ellipsoidal spiral coordinatesystem. A display may correspond to a gather of image points with thesame lateral location and different depths. Initially the parameters ofthese spirals (e.g., elevation, maximum zenith angle, segment area,etc.) may be different. Thus, there may be several different spiralrepresentations of the computed data (e.g., image function),corresponding to different depths of the gather nodes. In addition,there may be a visualization spiral to which parameters may be assigned.The maximum zenith angle of the visualization spiral may be the largestmaximum zenith angle for all gather nodes at different depths. Toconstruct a common cylindrical display 300 and/or 400 for all imagepoints of the gather, the computed data may be regularized or normalizedto fit the nodes of the visualization spiral. Regularization may becarried out for each individual node in depth, and may include, forexample, two stages. First, gridding may be performed for the input dataat the control points with specific angular locations, corresponding tothe given input or non-regularized spiral. The input spirals may bedifferent for different depths. For the reflection subsystem gather, themaximum zenith angle (e.g., the maximum opening angle) typicallydecreases with depth. Theoretically, the maximum opening angle mayvanish at infinite depth. After gridding, the values of the imagefunction may be determined or scanned at any point of the spherical orellipsoidal surface. In one embodiment, the values of the image functionmay be “read” at the nodes of the visualization spiral (e.g., nodes 830of the continuous curved three-dimensional surface 835 of FIG. 8C). Thevisualization spiral may be the same for all nodes (e.g., all depthindices) of the gather. Alternatively, the visualization spiral may bedifferently shaped or have different nodal arrangements or ordering atone or more different nodes at depths of the gather, for example, toaccommodate for inconsistent data.

A gridding technique may be performed independently for each sphericalor ellipsoidal surface (e.g., of FIGS. 1 and 2) (e.g., for each node indepth). Alternatively, a gridding technique may be performedsimultaneously, one time, or according to the same parameters, for thewhole gather or portion thereof. The latter method may increase thecomputational complexity of gridding, but may provide a more continuousvertical distribution of the image function.

In one embodiment, the visualization spiral data set for each node ofthe gather may be mapped to a continuous distribution, generating aspherical or ellipsoidal display. The spherical or ellipsoidal displaygenerated for each vertical node of the gather may be projected,flattened or expanded, into a planar circular, elliptic orelliptic-like, or otherwise planar 2D region. In one embodiment, thepolar radius of a point on the curved 2D region may be equal orproportional to the zenith angle on the spherical display. The azimuthvalues of the points on the 3D spherical and flattened 2D displays aretypically the same.

Note that for the reflection subsystem representing the offset magnitudeand the offset azimuth, each single-node of the gather may be originallydefined on a planar surface, and need not be projected or expanded ontoa plane.

A set of planar displays (e.g., circular or planar curved-linenon-circular) created for the nodes of the gather with the same laterallocation and different vertical locations may form each cylindricaldisplay 300 and/or 400.

In one embodiment, reflection and directional subsystem cylindricaldisplays 300 and 400 may include horizontal cross sections 320 and 420and vertical cross sections 330 and 430 thereof, respectively.Horizontal cross sections 320 and 420 may be normal to axes 310 and 410,respectively, and vertical cross sections 330 and 430 may be parallel toaxes 310 and 410, respectively. Each horizontal cross section 320 and420 may be, for example, a projected, flattened, or reduced dimensionalrepresentation of reflectivity and directional spherical displays 100and 200, respectively. Horizontal cross sections 320 and 420 may includeregions 365 and 465. Regions 365 and 465 may be, for example, projected,flattened, or reduced dimensional representation of reflectivity vs.reflection and directional component angles of regions 165 and 265,respectively, for an image point. A color map including a plurality ofcolors may correspond to a range of the image function values. In FIGS.3 and 4, different colors are represented by differently shaded regions365 and 465, respectively. The color map may be the same color map asused for the spherical displays in FIGS. 1 and 2 for the correspondingimage point of the gather. Alternatively, a different color map may beused. Other methods of displaying image function values may be used,such as e.g., grayscale, brightness, luminosity, cross-hatching, etc.

A user may view displays on a graphical user interface (e.g., graphicaluser interface 182 a and/or 182 b of FIG. 7).

Horizontal cross section 320 and 420 may correspond to a specific imagepoint of the gather (e.g., with a specific vertical location).Horizontal cross sections 320 and 420 may show the reflectivity functionin, for example, a range of or all spatial directions.

Vertical cross sections 330 and 430 may include information related to agather of points, for a range of zenith angles, and for a specific orsingle azimuth value.

The cylindrical radius 332 (e.g., of the meridian gather), correspondingto the amplitude of an image function, may vary along the length of thevertical axis 310. For example, in vertical cross sections 330, thecylindrical radius 332 may decay or reduce as the depth or axis 310coordinate increases. Thus, for a limited acquisition area or a limiteddistance between the source and the receiver, the maximum possibleopening angle between the incident and the reflected rays typicallydecrease with increasing depth and, for example, a zenith angleapproaching an infinite depth may vanish (e.g., or approach a negligentor zero value). For the directional subsystem, along vertical crosssection 430, the maximum zenith angle may also decay with depth.

Cylindrical radius 332, axes 310 and 410, image function, etc. may beviewed on the cylindrical displays 300 and 400, respectively, or asseparate data points or graphs. In another embodiment, this data may behidden or revealed according to a selection by a user.

Although displays of three-dimensional bodies shaped as cylinders aredescribed herein, it may be appreciated by the one skilled in the artthat differently shaped three-dimensional bodies may be used toconcurrently represents data associated with a plurality of imagepoints. For example, a plurality of spherical surface displays, eachrepresenting data associated with a single point, may be assembled intoshapes other than cylinders.

In one embodiment, each of the spherical surface displays may beresized, e.g., to a different graded size (monotonically increasing ordecreasing in size). Each resized spherical surface may be nestedaccording to its size to form a solid sphere. For example, the smallestspherical surface may be an interior surface of the solid sphere and alargest spherical surface may be an exterior of the solid sphere). Theresizing of each spherical surface (e.g., enlarging or decreasing) maybe considered a projection map onto a differently sized sphericalsurface (e.g., larger and smaller, respectively). In one embodiment,there may be a minimum size for the smallest spherical surface, e.g.,for that surface to contain and display (with proper resolution as to beuseful to a user) the data associated with a single seismic image point.In this embodiment, the spherical three-dimensional body or solid may behollow, e.g., having no data at a radii less than the radius of thesmallest spherical surface. In other embodiments, the smallest sphericalsurface may be a point and the spherical body or solid may not behollow. In such embodiments, the smallest one or more spherical surfacemay be used as placeholders. Since the smallest spherical surface aretypically too small to visualize data associated therewith with properresolution, a separate display may provide this data (e.g., as a fullsize spherical surface in a “pop-up” window).

Other shapes for the (e.g., solid) three-dimensional body may include ahyperboloid, ellipsoid, and polyhedron, e.g., which may be assembledfrom a plurality of nested hyperbolic surfaces, elliptical surfaces, andpolyhedron surfaces, respectively, (e.g., each of differently gradedsize).

A three-dimensional body may define a solid object in athree-dimensional coordinate space. A three-dimensional surface maydefine a two-dimensional manifold, e.g., in a three-dimensionalcoordinate space. Other definitions of surfaces and bodies may be used.Spherical displays 100 and 200 of FIGS. 1 and 2, respectively, aretypically three-dimensional surfaces and cylindrical displays 300 and400 of FIGS. 3 and 4, respectively, are typically three-dimensionalbody.

Example of Workflow

Reference is again made to FIGS. 5 and 6, which are flowcharts ofmethods, according to embodiments of the invention. For example, FIG. 5describes operations for generating and displaying spherical displays(e.g., shown in FIGS. 1 and 2) and cylindrical displays (e.g., shown inFIGS. 3 and 4), according to one embodiment. Other embodiments may beused to create such displays.

The flowcharts describe an example of workflow for data processing anddata visualization, for example, using spherical and cylindricaldisplays. It may be appreciated by the one skilled in the art that thisworkflow is only an example, among a variety of other possible sequencesof operations, and this workflow example does not limit the invention.

Referring to FIG. 5, in operation 500, a seismic acquisition system(e.g., system 700 of FIG. 7) may collect seismic data. Other data, suchas medical imaging data or industrial imaging data, may be collected,processed, and displayed.

In operation 510, a processor (e.g., processor 140 of FIG. 7) maygenerate a discrete data set (e.g., discrete data set 815 or 825 ofFIGS. 8A and 8B) representing an image function for an image point. Thediscrete data set may include seismic data gathers traced from theacquired seismic data. The seismic data gathers may be, for example,discrete azimuthally dependent angle and/or offset domain gathers forthe directional and/or reflection subsystems. Other data or types ofgathers may be used.

In operation 520, the processor may normalize (or grid) an irregulardata set (e.g., irregular discrete data set 815 of FIG. 8A) (e.g., usingthe first map 860). For example, a computational mesh (e.g.,computational mesh 822) may be generated along which data nodes (e.g.,nodes 820) or control points of the discrete data set may be regularlyor evenly spaced. This operation may be skipped if the nodes or controlpoints of the discrete data set are already regularly or evenly spaced.

In operation 530, the processor may, for each gather node, and for eachsubsystem (e.g., the directional and reflection subsystems), apply aninterpolation technique (e.g., using the second map 865), to obtain thevalues of the image function between the regularly spaced nodes of themesh (e.g., on the continuous curved three-dimensional surface 835).

In operation 540, for selected gather nodes, and for each of thesubsystems, the processor may plot the image function on a spherical orotherwise curved surface (e.g., spherical displays 100 and 200, of FIGS.1 and 2, respectively). The displayed color map or other map may includea continuous distribution of the image function through the spherical orotherwise curved surface (e.g., continuous curved three-dimensionalsurface 835).

In operation 550, for each gather node, the processor may (e.g., usingthe third map 870), project, flatten, or expand the image date definedon the spherical surface or otherwise curved surface, onto a planarsurface 845 (e.g., a circular or other disk) such as a circular ornon-circular planar disk.

In operation 560, for each of the two subsystems, the processor mayassemble planar disks having the projected image date (e.g., using thefourth map 875). The data defined with planar surfaces for each gathernode may be combined to form a three-dimensional body (e.g.,three-dimensional body 855) such as a circular or non-circular cylinder.

In operation 570, the processor may display the three-dimensional body(e.g., cylindrical displays 300 and 400, of FIGS. 3 and 4,respectively).

In operation 580, the processor may generate axial cross sections (e.g.,330 and 430 of FIGS. 3 and 4, respectively) of the three-dimensionalbody (e.g., of cylindrical displays 300 and 400, respectively) forarbitrary azimuths selected by a user. Each axial cross section mayrepresent a single-azimuth image gather including multiple zenith angles(e.g., or offset magnitudes) within a given range, and multiple gathernodes in depth.

Other operations or series of operations may be used.

Referring to FIG. 6, in operation 600, a first display (e.g., display180 a of FIG. 7) may show a standard display of image points (e.g.,seismic data) as a visualization of a geophysical region. The firstdisplay may be a visualization of an image function vs. location. Forexample, each point may be defined by three components, such as,Cartesian components of the point, and a color corresponding to afunction value at each point. While specific displays (e.g. FIGS. 1-4)and data structures are described, other displays and data structuresmay be used with embodiments of the invention.

In operation 610, an input device may receive point selected orindicated by a user. The user may indicate (e.g., by clicking orhighlighting) the physical location of one image point or a plurality ofimage points (e.g., determined to be located along a line in a physicalspace, e.g. a vertical line) of the standard display to be displayed asa sphere or cylinder. Points may be “determined” to be located insteadof “actually” located, since geophysical simulations typically have lessthan full (100%) accuracy. In response to such an indication, a displaymay display (e.g., on the graphical user interface) a representation ofthe indicated data to a user.

In operation 620, in response to receiving indications of one or moreseismic image points from a user, a second display (e.g., display 180 bof FIG. 7) may display (e.g., on graphical user interface 182 b), one ora plurality (e.g., a gather) of image points, of one or more componentsof data in addition to the data displayed in operation 600 (e.g., ongraphical user interface 182 a). The second display may be avisualization of the image function vs. additional component, such as,an orientation on the spherical display, or a location (e.g. vertical)and orientation on the cylindrical display. When data for one imagepoint is displayed, the display may include spherical displays 100and/or 200 of FIGS. 1 and 2. When data for a plurality of image point orseismic image gather (e.g., corresponding to a line or curve in aphysical space) is displayed, the display may include cylindricaldisplays 300 and 400 of FIGS. 3 and 4. The additional component datadisplays (e.g., on graphical user interface 182 b) may be displayed withor adjacent to a standard display or visualization of a geophysicalregion (e.g., on graphical user interface 182 a). While two displays ormonitors are shown, this is by way of example only, and only one or morethan two monitors may be used, and further only one type of data need bedisplayed at one time. For example, a “standard” 3D display need not bedisplayed, but rather specific components.

In one embodiment, the cylindrical display (e.g., 300 and/or 400, ofFIGS. 3 and 4) may be viewed as a partially hollowed structure.Alternatively, the display may be (e.g., initially) solid and whenoperated on or manipulated by a user, may become transparent to revealdata not shown on the outside surface of the solid structure. Forexample, only one planar disk may be displayed at a time. The otherdisks may be hidden so that they do not obstruct a view of the singledisplayed disk. In other embodiments, two or more disks may besimultaneously displayed. For example, a user may indicate a point(e.g., along the line 880) on the cylinder and in response to theindication, the display may display the disk having that point. Datacorresponding to the hidden disks, e.g., the maximum cylindrical radius332 (e.g., of the meridian gather), corresponding to the maximum zenithangle of the image function of a disk varying along the length of thevertical axis 310, may be displayed. The cylindrical radius 332 of themeridian gather may be a simple indication of the image function ofdisks that are not displayed. A user may for example compare thedetailed image function of the single displayed disk to the simplifiedrepresentation of the image function (e.g., a maximum cylindrical radius332 representing maximum zenith angle of the meridian gather) of thehidden disks, e.g., to identify trends or compare image function orcomponent values among disks.

Other portions of the displays may be hidden or revealed and the hiddenor revealed structures may be changeable, e.g., controlled at least inpart by a user operating an input device. For example, when displayingcylindrical display (e.g., 300 and/or 400, of FIGS. 3 and 4), a user mayrequest (e.g., by right clicking or selecting an appropriate icon) tohide or to reveal axes 310 and/or 340, planar nodes 840 (e.g., e.g.,overlaid on corresponding planar disks), and/or cylindrical nodes 850(e.g., e.g., overlaid on the corresponding cylindrical displays). Forexample, when displaying spherical display (e.g., 100 and/or 200, ofFIGS. 1 and 2), a user may request to hide or to reveal computationalmesh 822, or an output mesh thereof, e.g., spiraling nodes 830, etc. Theuser may request additional information to reveal the physical space(e.g., by color, highlighting, arrow pointers, etc.) to which a selectedone or more image points are determined to be located (e.g., on thefirst display of operation 600). Additional data may be shown, e.g., thenumber of iterations used to generate a computational mesh 822 or avisualization of the stages of the recursive construction (e.g., from anfirst stage polygon having a relatively few number of faces to a laststage polygon having a relatively large number of faces and closelyapproximating a sphere to within a predetermined threshold of errorcompared to the next iteration). Other data, such as, maximumcylindrical radius 332 of a line of points such as the meridian gather,color maps, axes 310 and 410, locations of image points on thecylindrical display and/or in a physical space, image function values,color maps, etc. may be viewed on the display or as separate data pointsor graphs. This data may be hidden or revealed according to a selectionby a user.

In one embodiment, a cylindrical display (e.g., 300 and/or 400, of FIGS.3 and 4) may display components of image data corresponding to twosubsystems (e.g., reflection angle or offset and direction). Thecylindrical display may include multiple stacked planar disks, eachflattened from a spherical display (e.g., 100 and/or 200, of FIGS. 1 and2), each spherical display representing an image function for adifferent point of an image gather. The cylindrical display may thuscorrespond to an image gather. Displays other than a cylindrical displaymay be used, such as cones, pyramids, funnels, conical or cylindricalportions or sections, etc.

The user may, for example, analyze the pair of reflection (e.g., oroffset) and directional displays corresponding to the same point. A pairof displays including a directional display having a single or narrowrange of values and a reflection angle (e.g., or offset) display havingmultiple or a wide range of values the image point may indicate that theimage point lies on a fault line.

Thus, operations 610 to 620 may be repeated, for example, as a userindicate other image points (e.g., scanning image points), to viewcorresponding pairs of directional and reflection angle (e.g., oroffset) displays to discover the locations of image points that fall onfault lines.

Other benefits may be realized.

Other operations or series of operations may be used.

Example of a System

Reference is made to FIG. 7, which is a schematic illustration of asystem, including a transmitter, a receiver and a computing system inaccordance with an embodiment of the present invention. System 700 maybe used, for example, to display image data, such as, for example,seismic data, in a spherical displays (e.g., of FIGS. 1 and 2) and/orcylindrical displays (e.g., of FIGS. 3 and 4) and/or displays of astandard or physical space. Data other than seismic data may be usedwith embodiments of the present invention. System 700 may performembodiments of any of the methods discussed herein, and/or otheroperations or computations. System 700 may, for example, display astandard display or visualization of physical space (e.g., a geophysicalspace, a medical image), and in addition or as an alternate option for auser, displays such as but not limited to spherical or cylindricaldisplays, which may display details or additional component data whichmay have been used to build the standard visualization. System 700 mayinclude for example a transmitter 110, a receiver 120, a computingsystem 130, and one or more displays 180 a and 180 b. In a system havingmore than one monitor or display, a standard display or visualization ofa physical space may be displayed on one display (e.g., display 180 a),and spherical or cylindrical displays displaying details or additionalcomponent data may appear on a second display (e.g., display 180 b).However, one display may be used for displaying one or more types ofinformation (e.g., a standard display and/or a display includingadditional component data). Transmitter 110 may output any suitablesignals, or generate incident signal(s). For example, a series of sonicor seismic energy rays or waves may be emitted from each of multiplelocations. Receiver 120 may accept reflected signal(s) that correspondor relate to incident signals, sent by transmitter 110. In the case ofimaging in other areas, e.g., medical imaging, transmitter 110 mayoutput energy such as ultrasound, magnetic, x-ray, or other suitableenergy.

Computing system 130 may include, for example, processor 140, memory 150and software 160. Processor 140 may process data, for example, raw datareceived from receiver 120. Memory 150 may store data, for example, rawor processed seismic data. Operations executed according to embodimentsof the invention, such as for example, mapping, projecting,interpolating, gridding, generating a computational mesh, estimating,approximating, displaying, etc. may be at least partially executed,operated or calculated, for example, by an operator (e.g., implementedin software 160). Other units or processors may perform such operations,or other operations according to embodiments of the present invention.

Displays 180 a and/or 180 b (e.g., such as monitors or screens) maydisplay to a user or viewer spherical or cylindrical displaysrepresenting data from transmitter 110, receiver 120, computing system130, or any other suitable systems, devices, or programs, for example,an imaging program or software, or a transmitter or receiver trackingdevice. Displays 180 a and/or 180 b may include one or more inputs fordisplaying data from multiple data sources. The system may includemultiple displays. Displays 180 a and/or 180 b may display imagesproduced from data. For example, displays 180 a and/or 180 b may displayrepresentations or visualizations of seismic or other imaging data, forexample, angle dependent CIGs, processes according to embodimentsdescribed herein.

Displays 180 a and 180 b may include for example one or more graphicaluser interfaces 182 a and 182 b, respectively. The displays may, inresponse to an indication, from a user operating the input device 170,of one or more seismic image points or a line of a physical space,display the graphical user interfaces 182 a and/or 182 b. The displaymay, for example, display direction data, data for a plurality ofdirections, and/or an image function vs. the direction, associated withthe indicated seismic image point. The graphical user interfaces 182 aand/or 182 b may enable a user to view the results of their input (e.g.,operating the input device 170) or interact with the displays forselecting points, requesting additional data, setting graphicalparameters, selecting to hide or reveal graphical structures, etc.

Computing system 130 may include, for example, one or more inputdevice(s) 170 (e.g., such as a keyboard and/or mouse) for receivingcommand, selections, or signals (e.g., from a user or other externaldevice).

Computing system 130 may include, for example, any suitable processingsystem, computing system, computing device, processing device, computer,processor, and the like, and may be implemented using any suitablecombination of hardware and/or software.

Processor 140 may include, for example, one or more processors,controllers or central processing units (“CPUs”). Software 160 may bestored, for example, all or in part, in memory 150. Software 160 mayinclude any suitable software, for example, for processing or imagingaccording to embodiments of the present invention. Processor 140 mayoperate at least partially based on instructions in software 160.Embodiments of the invention may include a computer readable storagemedium, such as for example a memory, a disk drive, or a “disk-on-key”,including instructions which when executed by a processor or controller,carry out methods disclosed herein, or cause the processor to carry outsuch methods. Software 160 may be stored on the computer readablestorage medium.

System 700 may, for example, display images of target surfaces, forexample, using software 160 and/or processor 140, or other componentssuch as dedicated image or signal processors. Such displays may be usedand analyzed for example, for determining the locations of image pointsthat lie on faults or other geological discontinuities or points ofinterest.

System 700 may, for example, display a seismic image data point. Thespatial features of the seismic data point (e.g., a zenith and azimuthangles, a relative location, and/or other spatial coordinates within alarger subsurface region) may be represented in a standard display of aconventional (e.g., Cartesian or polar) coordinate system.

Processor 140 may, for example, compute, from a wide-azimuth data set, adiscrete data set (e.g., 815 or 825 of FIGS. 8A and 8B, respectively)associated with an image function at a seismic image point in athree-dimensional coordinate system. The processor 140 may map (e.g., byinterpolation or gridding) the discrete data set onto a continuouscurved three-dimensional surface (e.g., 835 of FIG. 8C).

Processor 140 may generate a computational mesh (e.g., according tooperation 520 of FIG. 5). The computational mesh (e.g., computationalmesh 822 of FIG. 8B) may include regularly spaced nodes (e.g., nodes820). Processor 140 may grid (e.g., or interpolate) the input data ontothe computational mesh (e.g., according to operation 530 of FIG. 5). Forexample, processor 140 may map the discrete data set onto the continuouscurved three-dimensional surface, for example, by estimating the valuesof the image function at the nodes of the computational mesh and atpoints between the regularly space nodes of the computational mesh.Processor 140 may generate the computational mesh starting from aprimary polyhedron inscribed into a sphere and then splitting itsspherical triangles in a recursive manner, as described in the previoussections.

Other relationships or maps between the spatial coordinates and thecontinuous curved three-dimensional surface may be used.

Displays 180 a and/or 180 b may display the continuous curvedthree-dimensional surface as a sphere, an ellipsoid, a spherical cap,and an ellipsoidal cap, or otherwise curved 3D surface (e.g., such asspherical displays 100 and/or 200 of FIGS. 1 and 2, respectively).

Processor 140 may, for example, project the mapped data set onto acontinuous curved two-dimensional surface (e.g., a planar disk).Displays 180 a and/or 180 b may display the projected data as a planardisk. The outer boundary of the planar disk may be, for example, aclosed curve, including a circle, an ellipse, or another closed curve,or an otherwise 2D surface (e.g., such as displays of planar surface 845of FIG. 8D). The radii of the points on the planar disk may, forexample, correspond to zenith angles of the discrete data set or tooffset magnitudes of the discrete data set.

Processor 140 may, for example, assemble the plurality of continuousplanar surfaces into a three-dimensional body (e.g., 855 of FIG. 8E).Displays 180 a and/or 180 b may display the three-dimensional body as acylinder, a cone, a truncated cone, or otherwise curved 3D body (e.g.,such as cylindrical displays 300 and/or 400 of FIGS. 3 and 4,respectively).

The arguments of the image function represented on the continuous curvedthree-dimensional surface may include, for example, direction angles,reflection angles, offsets (e.g., defined by offset magnitudes andazimuths) of, e.g., wide-azimuth data collected during geophysicalexploration, and/or other or additional components not typically shownon a standard display. Although these additional components may becombined or processed to provide spatial features of the image point,each of the additional components typically do not show a location ofthe image point in a geophysical space.

System 700 may, for example, display a seismic image gather. Processor140 may generate, from a wide-azimuth data set, a plurality of discretedata sets (e.g., 815 or 825 of FIGS. 8A and 8B, respectively). Eachdiscrete data set may be associated with an image function of one of aplurality of seismic image data points determined to be located along aline in a physical space. Each of the plurality of seismic image datapoints determined to be located along a line in a physical space maycorrespond to a seismic image gather representing a plurality of zenithangles and a single azimuth angle.

Processor 140 may grid each of the discrete data sets onto a continuouscurved three-dimensional surface. Processor 140 may project each of thegridded data sets onto a continuous curved two-dimensional surface.Processor 140 may assemble the plurality of continuous curvedtwo-dimensional surface into a three-dimensional body. Processor 140 mayassemble the plurality of continuous curved two-dimensional surfacesinto a three-dimensional body along an axis (e.g., line 880 of FIG. 8E)of symmetry of the body.

Displays 180 a and/or 180 b may display the three-dimensional body. Thethree-dimensional body for concurrently representing data associatedwith the line of points. The three-dimensional body may be, e.g., such acylinder (e.g., such as cylindrical displays 300 and/or 400 of FIGS. 3and 4, respectively), a cone, a funnel, or a pyramid.

An input device (e.g., input device 170 of FIG. 7) may receiveindications from a user of a seismic image point or a line (e.g., ofseismic image points) in a physical space. The display may, in responseto the indication of the seismic image point from a user operating theinput device, display a graphical user interface (e.g., graphical userinterface 182 b of FIG. 7) comprising projected data derived from thediscrete data set associated with the indicated seismic image point. Thedisplay may, in response to an indication of the line from a useroperating the input device, display a graphical user interface (e.g.,graphical user interface 182 b) including the three-dimensional bodyderived from the discrete data set associated with the plurality ofseismic image data points determined to be located along the indicatedline.

One embodiment of the invention may include displaying wide-azimuthseismic image data mapped from a first coordinate system to a reduceddimensional coordinate system for reducing the dimensionality of thedata set. The reduced dimensional coordinate system may be for example,as described in U.S. patent application Ser. No. 11/798,996. A displaymay include a spiraling geometry. According to one embodiment, thereduced dimensional data may be displayed on spherical surfaces for thereflection angle (FIG. 1) and directional (FIG. 2) subsystems of theLAD. These displays may be projected and assembled to form reduceddimensional cylindrical displays in the reduced dimensional coordinatespace.

The foregoing description of the embodiments of the invention has beenpresented for the purposes of illustration and description. It is notintended to be exhaustive or to limit the invention to the precise formdisclosed. It should be appreciated by persons skilled in the art thatmany modifications, variations, substitutions, changes, and equivalentsare possible in light of the above teaching. It is, therefore, to beunderstood that the appended claims are intended to cover all suchmodifications and changes as fall within the true spirit of theinvention.

1. A method for displaying data representing a single seismic image datapoint of subsurface geological structures, the method comprising:computing, from a wide-azimuth data set, a discrete data set includingmultiple azimuths associated with an image function at the singleseismic image point; mapping the discrete data set onto a continuouscurved three-dimensional surface; displaying the mapping of the discretedata set computed from wide-azimuth data representing the single seismicimage data point of the subsurface geological structures on thecontinuous curved three-dimensional surface.
 2. The method of claim 1,wherein the discrete data set is mapped from a first coordinate systemto the continuous curved three-dimensional surface in a reduceddimensional coordinate system to reduce the dimensionality of thediscrete data set.
 3. The method of claim 1, wherein the discrete dataset comprises zenith angles and azimuth angles, wherein each distinctpair of zenith and azimuth angles of the discrete data set correspondsto a point of the continuous curved three-dimensional surface.
 4. Themethod of claim 1 comprising defining a computational mesh withregularly spaced nodes on the continuous curved three-dimensionalsurface, and mapping the discrete data set onto the continuous curvedthree-dimensional surface by estimating the values of the image functionat the nodes of the computational mesh and at points between theregularly spaced nodes of the computational mesh.
 5. The method of claim1 comprising defining the discrete data set on discrete nodes of aspiraling geometry.
 6. The method of claim 5 comprising performing aspherical spiral discretization to define the location of input pointsby a normalized area swept by spiral coils or a normalized arc length ofthe spiraling geometry.
 7. The method of claim 1, wherein the continuouscurved three-dimensional surface is selected from the group consistingof: a sphere, an ellipsoid, a spherical cap, and an ellipsoidal cap. 8.The method of claim 1, wherein displaying comprises, in response to anindication of a location of a seismic image point by a user, displayingto the user the image function vs. the direction associated with theindicated seismic image point.
 9. The method of claim 1 comprising:projecting the mapped data set onto a continuous planar surface; anddisplaying the projection of wide-azimuth data on a planar diskrepresenting the single seismic image data point.
 10. The method ofclaim 1, comprising: generating a plurality of continuous planarsurfaces representing data associated with a plurality of respectiveseismic image data points determined to be located along a line in aphysical space; assembling the plurality of continuous planar surfacesinto a three-dimensional body; and displaying the three-dimensional bodyfor concurrently representing data associated with the line of points.11. A system for displaying data representing a single seismic imagedata point of subsurface geological structures, the system comprising: aprocessor configured to: compute, from a wide-azimuth data set, adiscrete data set including multiple azimuths associated with an imagefunction at the single seismic image data point, and map the discretedata set onto a continuous curved three-dimensional surface; and adisplay to display the mapped discrete data set computed fromwide-azimuth data representing the single seismic image data point ofthe subsurface geological structures on the continuous curvedthree-dimensional surface.
 12. The system of claim 11, wherein theprocessor is configured to map the discrete data set from a firstcoordinate system to the continuous curved three-dimensional surface ina reduced dimensional coordinate system to reduce the dimensionality ofthe discrete data set.
 13. The system of claim 11, wherein the processoris configured to compute the discrete data set to include zenith anglesand azimuth angles, such that each distinct pair of zenith and azimuthangles of the discrete data set corresponds to a point of the continuouscurved three-dimensional surface.
 14. The system of claim 11, whereinthe processor is configured to define a computational mesh withregularly spaced nodes on the continuous curved three-dimensionalsurface, and map the discrete data set onto the continuous curvedthree-dimensional surface by estimating the values of the image functionat the nodes of the computational mesh and at points between theregularly spaced nodes of the computational mesh.
 15. The system ofclaim 11, wherein the processor is configured to define the discretedata set on discrete nodes of a spiraling geometry.
 16. The system ofclaim 15, wherein the processor is configured to perform a sphericalspiral discretization to define the location of input points by anormalized area swept by spiral coils or a normalized arc length of thespiraling geometry.
 17. The system of claim 11, wherein the continuouscurved three-dimensional surface is selected from the group consistingof: a sphere, an ellipsoid, a spherical cap, and an ellipsoidal cap. 18.The system of claim 11, wherein the processor is configured to displayby, in response to an indication of a location of a seismic image pointby a user, displaying to the user the image function vs. the directionassociated with the indicated seismic image point.
 19. The system ofclaim 11, wherein the processor is configured to project the mapped dataset onto a continuous planar surface and display the projection ofwide-azimuth data on a planar disk representing the single seismic imagedata point.
 20. The system of claim 11, wherein the processor isconfigured to: generate a plurality of continuous planar surfacesrepresenting data associated with a plurality of respective seismicimage data points determined to be located along a line in a physicalspace, assemble the plurality of continuous planar surfaces into athree-dimensional body, and display the three-dimensional body forconcurrently representing data associated with the line of points.